Cauchy Sequences

What is slightly annoying for the mathematician (in theory and in praxis) is that we refer to the limit of a sequence in the definition of a convergent sequence when that limit may not be known at all. In fact, more often then not it is quite hard to determine the actual limit of a sequence.

We would prefer to have a definition which only includes the known elements of the particular sequence in question and does not rely on the unknown limit. Therefore, we will introduce the following definition:

Definition: Cauchy Sequence

Let be a sequence of real (or complex) numbers. We say that the sequence satisfies the Cauchy criterion (or simply is Cauchy) if for each > 0 there is an integer N > 0 such that if j, k > N then | a_j - a_k | < .

This definition states precisely what it means for the elements of a sequence to get closer together, and to stay close together. Of course, we want to know what the relation between Cauchy sequences and convergent sequences is.

Theorem: Completeness Theorem in R

Let be a Cauchy sequence of real numbers. Then the sequence is bounded.
Let be a sequence of real numbers. The sequence is Cauchy if and only if it converges to some limit a.

Thus, by considering Cauchy sequences instead of convergent sequences we do not need to refer to the unknown limit of a sequence, and in effect both concepts are the same.

Note that this proposition not true if we consider only rational numbers. For example, the sequence 1, 1.4, 1.41, 1.414, ... (convergent to the square root of 2) is Cauchy, but does not converge (in the rational numbers). Therefore, the rational numbers are not complete, in the sense that not every Cauchy sequence of rational numbers converges to a rational number.

Hence, the proof will have to use that property which distinguishes the reals from the rations: the least upper bound property.

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